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A333089 a(n) is the denominator of Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)).

%I A333089 #45 May 08 2021 08:33:31
%S A333089 1,18,960,76440,37437400,157024707840,2777798704721040,
%T A333089 4169982785629476816,331259342780844858796416,
%U A333089 743322803326470921519628462800,163037651356772148158514292729628880,187555796967791569325602741834073910082560,3838658658324493911932517275499048601188128008800
%N A333089 a(n) is the denominator of Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)).
%C A333089 The numerators are given in A333088.
%C A333089 Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n)) is a fraction for n > 0.
%C A333089 Sum_{i > 0} 1/Fibonacci(i)^2, i.e., the n = 0 case, is known to be transcendental. See A105393.
%C A333089 Sum_{i > 0} 1/(Fibonacci(i)*Fibonacci(i+2n-1)) is an irrational number for n > 0. See for instance A290565 (n = 1).
%H A333089 A.H.M. Smeets, <a href="/A333089/b333089.txt">Table of n, a(n) for n = 1..62</a>
%H A333089 Brother Alfred Brousseau, <a href="https://www.mathstat.dal.ca/FQ/Scanned/7-2/brousseau1.pdf">Summation of Infinite Fibonacci Series</a>, The Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 143-168. See (5) and (6) p. 148.
%H A333089 Stanley Rabinowitz, <a href="https://www.mathstat.dal.ca/FQ/Scanned/37-2/rabinowitz1.pdf">Algorithmic summation of reciprocals of products of Fibonacci numbers</a>, The Fibonacci Quarterly, Vol. 37 (1999), pp. 122-127. See (23) and (25) p. 5.
%F A333089 a(n) = denominator of (1/Fibonacci(2n)) * Sum_{0 < i <= n} 1/(Fibonacci(2i-1)*Fibonacci(2i)).
%e A333089 These infinite sums begin: 1, 7/18, 143/960, ...
%t A333089 a[n_] := Denominator[Sum[1/(Fibonacci[2i-1]*Fibonacci[2i]),{i,1,n}] / Fibonacci[2n]]; Array[a, 13] (* _Amiram Eldar_, Mar 10 2020 *)
%o A333089 (PARI) a(n) = denominator(sum(i=1, n, 1/(fibonacci(2*i-1)*fibonacci(2*i)))/ fibonacci(2*n)); \\ _Michel Marcus_, Mar 10 2020
%o A333089 (Python)
%o A333089 from math import gcd
%o A333089 f0, f1, snum, sden, n = 1, 1, 0, 1, 0
%o A333089 while n < 13:
%o A333089     snum, sden, n = f0*f1*snum+sden, sden*f0*f1, n+1
%o A333089     d = gcd(snum, sden*f0)
%o A333089     print(n, sden*f0//d)
%o A333089     f0, f1 = 2*f0+f1, f0+f1 # _A.H.M. Smeets_, May 16 2020
%Y A333089 Cf. A105393, A290565, A333088 (numerator).
%K A333089 nonn,frac
%O A333089 1,2
%A A333089 _A.H.M. Smeets_, Mar 07 2020